Download A self-consistent derivation of ion drag and Joule heating

Annales Geophysicae, 23, 3313–3322, 2005
SRef-ID: 1432-0576/ag/2005-23-3313
© European Geosciences Union 2005
Annales
Geophysicae
A self-consistent derivation of ion drag and Joule heating for
atmospheric dynamics in the thermosphere
Xun Zhu, E. R. Talaat, J. B. H. Baker, and J.-H. Yee
The Johns Hopkins University, Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723-6099, USA
Received: 21 May 2005 – Revised: 11 September 2005 – Accepted: 12 September 2005 – Published: 30 November 2005
Abstract. The thermosphere is subject to additional electric and magnetic forces, not important in the middle and
lower atmosphere, due to its partially ionized atmosphere.
The effects of charged particles on the neutral atmospheric
dynamics are often parameterized by ion drag in the momentum equations and Joule heating in the energy equation.
Presented in this paper are a set of more accurate parameterizations for the ion drag and Joule heating for the neutral
atmosphere that are functions of the difference between bulk
ion velocity and neutral wind. The parameterized expressions also depend on the magnetic field, the Pedersen and
Hall conductivities, and the ratio of the ion cyclotron frequency to the ion-neutral collision frequency. The formal relationship between the electromagnetic energy, atmospheric
kinetic energy, and Joule heating is illustrated through the
conversion terms between these three types of energy. It is
shown that there will always be an accompanying conversion of kinetic energy into Joule heating when electromagnetic energy is generated through the dynamo mechanism of
the atmospheric neutral wind. Likewise, electromagnetic energy cannot be fully converted into kinetic energy without
producing Joule heating in the thermosphere.
Keywords. Meteorology and atmospheric dynamics (Thermospheric dynamics) – Ionosphere (Ionosphere-atmosphere
interactions; Modeling and forecasting)
1 Introduction
Thermospheric dynamics is affected by charged particles
because the atmosphere is partially ionized. Atmospheric
flows consisting of major neutral species and minor charged
particles will differ from purely neutral flows because the
charged particles are subject to additional electric and magnetic forces. Collisions between atmospheric neutral species
and charged particles will yield a momentum exchange term
Correspondence to: X. Zhu
([email protected])
in the momentum equations (ion drag), and an energy conversion term in the thermal energy equation (Joule heating)
actualized by the neutral wind. Though the concentrations of
the charged particles are far less than those of the neutral atmosphere in the thermosphere below ∼1000 km (e.g., Kelley,
1989, Appendix B) they impose a non-negligible ion drag
and a Joule heating because the charged particles experiencing the electromagnetic forces may move in a significantly
different direction and magnitude than the neutral wind.
At least three different formulations can represent the ion
drag and Joule heating imposed on the neutral atmosphere
by charged particles: (i) the momentum sink and energy
dissipation of charged particles through microscopic collisions with neutral species when there exist velocity differences between the two; (ii) macroscopic Lorentz force and
Ohmic dissipation caused by electric currents and electromagnetic fields; or (iii) parameterized drag and heating that
are functions of the difference between the bulk ion velocity and neutral wind. The major difference between (i) and
(iii) is that velocities of ion and neutral species in (i) represent actual motions, are treated equally, and both need to be
solved self-consistently. On the other hand, the motion of the
bulk ion velocity in (iii) has been averaged over a gyrocycle,
is determined by the constant intrinsic magnetic field, and
is prescribed as model parameters for calculating the force
terms for the neutral wind. Depending on specific problems,
a particular formulation could be more useful for the purpose of an illustration or calculation. The microscopic description of interactions between neutral species and charged
particles through collisions leads to a solid foundation of the
transport phenomena associated with various physical quantities, including mass, momentum and energy (e.g., Gombosi, 1994; Schunk and Nagy, 2000). When measurements
of the electric and magnetic fields are available, the electromagnetic formalism becomes a useful tool for diagnosing
the thermospheric heat transfer (e.g., Lu et al., 1995, 1998;
Thayer, 1998). On the other hand, the parameterized ion drag
that linearly depends on the difference between the bulk ion
velocity and neutral wind provides both an insight into the
3314
X. Zhu et al.: A derivation of ion drag and Joule heating
physics and a method of straightforward implementation of
the ionospheric momentum source in numerical models (e.g.,
Dickinson et al., 1981). While these different approaches are
equally valid in describing essentially the same phenomena,
they are neither precisely the same nor always interchangeable. While formalisms of (i) and (ii) are readily available
and their differences described in the existing literature (e.g.,
St.-Maurice and Schunk, 1981), a set of parameterizations
for ion drag and Joule heating expressed as bulk ion-neutral
wind difference (iii) is presented here.
The purpose of this paper is to derive a set of parameterized expressions for the ion drag and Joule heating that is
more accurate than those shown in the literature when the
ion-neutral collisions become important. The parameterizations also possess an energetic consistency in their forms.
From an energetics perspective, ion drag and Joule heating
are associated with changes of three types of energy in the
system: the change of the electromagnetic energy caused by
electric currents, the change of kinetic energy caused by the
imposed drag force, and the change of the internal energy associated with the added heat. The effect of neutral wind on
Joule heating is discussed in parallel with analysis of energy
conversion terms. Such a detailed discussion has heretofore
been lacking in the literature. In Sect. 2, we review ion drag
in the macroscopic electromagnetic form and then derive explicit expressions parameterized by the bulk velocity difference. In Sect. 3, we present both formalisms for Joule heating. The energetic consistencies of the derived formalisms
and their implications are shown in Sect. 4. Finally, Sect. 5
summarizes the results.
2 Derivation of the parameterized ion drag terms in
momentum equations
In this paper, we focus on the thermospheric atmospheric
dynamics below ∼1000 km where the number density of
neutral particles is far greater than that of ions (e.g., Kelley, 1989, Appendix B). We mainly consider how the atmospheric motions of those major neutral species are affected
by the minor charged particles.
2.1
where
t
ns
ms
g
es
us
Ps
E
B
Ds
∂
Dt = ∂t +us ·∇
δM s
δt
At first glance, to quantitatively study the thermospheric
atmospheric motion subject to charged particles one needs
at least three momentum equations for three different types
of particles in the thermosphere: neutral particles (es =0),
single-charged ions (es =e), and electrons (es =−e). However, since neutral species remain the major components in
the thermosphere it is more sensible to focus on the dynamical equations for a neutral fluid with the parameterized effects of charged particles. An appropriate dynamical system
for major species is essential for thermospheric general circulation models since it provides a modeling frame for simulating the physics and chemistry of the rest of the minor
tracers, including charged particles (e.g., Dickinson et al.,
1975, 1981; Fuller-Rowell and Rees, 1980; Namgaladze et
al., 1990).
The momentum equations for the neutral components will
be Eq. (1) without the electric and magnetic forcing terms
(es =0). However, the collisional integral in the transport
Eq. (1) for a multi-species gas is too complicated for a
rigorous analytical solution. An alternative approach is
to start from the so-called magnetohydrodynamic equations
that combine the equations for individual species into a gas
mixture (e.g., Baumjohann and Treumann, 1996; Schunk and
Nagy, 2000). Because the collisional terms describe the internal transfer of momentum from one species to another,
the collisional terms cancel when the individual momentum
equations of Eq. (1) are summed. Thus, the general magnetohydrodynamic equation for the momentum conservation is
(Schunk and Nagy, 2000, p. 195)
Review of literature
ρ
To explicitly derive the bulk parameterization of the ion drag
in the momentum equation for the neutral atmosphere we
start from the general transport equation for an individual
species, s, (e.g., Schunk and Nagy, 2000, p. 54)
n s ms
Ds u s
δM s
+ ∇·Ps −ns ms g=ns es (E+us ×B)+
,
Dt
δt
s=1, 2, 3, . . . ,
(1)
= time
= species number density
= species mass
= acceleration due to gravity
= species charge (= 0, ±e with −e
being the electron charge)
= average velocity of species s
= partial pressure tensor for species s
= electric field
= magnetic field
= convective (or total) derivative
following us
= momentum source for species s due
to collisions
Du
+ ∇ · P − ρg = J × B,
Dt
(2)
where ρ is the total mass density, u is the average velocity,
P is the total pressure tensor, J is the total current density,
and D/Dt=∂/∂t + u·∇ is the convective derivative following the average velocity u. We have also neglected the force
term by the electric field, ρc E, due to the quasi-neutrality
approximation for the charge density (ρc =0) (Goldston and
Rutherford, 1995, Sect. 8.2). The notation in Eqs. (1) and (2)
mostly follows Schunk and Nagy (2000), to which readers
can refer to for more detailed definitions.
X. Zhu et al.: A derivation of ion drag and Joule heating
3315
In Eq. (2), ρ, u, and P are weighted by the total densities of both neutral species and charged particles whereas
the force term on the right-hand side associated with J is
weighted only by the density of charged particles. Therefore,
for a gas mixture where charged particles are considered minor tracers the left-hand side of Eq. (2) can be approximately
simplified by only retaining terms contributed from the major
species of the neutral atmosphere
ρn
Dn u n
+ ∇ · Pn − ρn g = J × B,
Dt
(3)
where all quantities on the left-hand side have the same definitions as in Eq. (2) and the subscript n refers to the neutral
atmosphere. The right-hand side of Eq. (3) is the ion drag
imposed on the neutral atmosphere as a result of the motions
of the charged particles.
The technique of scale analysis in simplifying a set of fluid
equations in the lower atmosphere proposes that a term of
much smaller magnitude in comparison with one or more remaining terms can be dropped (e.g., Haltiner and Williams,
1980, Ch. 3; Holton, 1992, Sect. 2.4). Two basic rules one
needs to follow when applying scale analysis to simplify
equations are: (i) small terms need to be dropped one by
one; and (ii) at least two terms need to be kept at the end of
simplification. The reason for having the first rule is that two
terms in an equation may form a near exact balance, so that
the sum of the two is much smaller than the rest of the terms,
although the absolute magnitude of the two individual terms
is much greater than the others. Elimination of those two
terms simultaneously may lead to an inappropriate equation
that represents the erroneous residual of two major physical
processes described by the two nearly balanced terms. The
second rule prevents the occurrence of the so-called “oneterm dominance” that always leads to a contradictory solution of a vanishing dominant term. Our simplification from
Eq. (2) to Eq. (3) for a gas mixture satisfies these two basic
rules since the dropped terms weighted by the minor tracers
are much smaller than the corresponding terms weighted by
the major species. Since the right-hand side of Eq. (2) had
already been weighted by the minor tracers it cannot be simplified further in Eq. (3) without a reference term that can be
compared to in magnitude.
Note that the left-hand side of Eq. (2) would be identical
to Eq. (3) if the momentum Eq. (1) were written for the neutral atmosphere (e=0), but the right-hand side would contain
the collisional term δM n /δt. δM n /δt describes the exact
microscopic transform of momentum (approach (i)) from the
charged particles to the neutrals whereas J ×B can be considered a macroscopic drag imposed on the neutral fluid (approach (ii)). Our resulting Eq. (3) based on the scale analysis
to a gas mixture is consistent with the recent approach by
Vasyliunas and Song (2005) who arrived at the same conclusion based on the scale analysis of a system of two separate
fluids.
2.2
Derivation of the parameterized ion drag
To calculate the ion drag term J ×B (=J ⊥ ×B because
J k ×B=0), we first note that the current density perpendicular to the magnetic field (J ⊥ ) is given by (e.g., Schunk and
Nagy, 2000, p. 131)
J ⊥ = σP (E ⊥ + un × B) + σH b × (E ⊥ + un × B),
(4)
where E ⊥ is the electric field perpendicular to B with b being the unit vector along B. σP and σH are the Pedersen and
Hall conductivities, respectively. σP and σH represent the
measures of the charged particle mobility parallel and perpendicular to E ⊥ , respectively. We have again adopted the
quasi-neutrality approximation for the charge density. The
perpendicular electric field E ⊥ under a strong intrinsic magnetic field, such as that in the Earth’s thermosphere, is related
to the cross-B current and can be approximately expressed as
(e.g., Schunk and Nagy, 2000, p. 130)
E ⊥ = −(ui × B) +
mi νi
(ui − un ),
e
(5)
where ui is the ion velocity and νi is the ion-neutral collision
frequency. Adoption of Eqs. (4) and (5) for J and E implies that the ion velocities in Eq. (1) are determined by the
strong intrinsic magnetic field. Note that ion and neutral velocities are treated equally as unknown dependent variables
in Eq. (1). However, using the bulk flow system of Eqs. (3),
(4) and (5), the bulk ion velocity is determined diagnostically
from Eq. (5). Substituting Eqs. (4) and (5) into the right-hand
side of Eq. (3) we obtain the ion drag term
F I −D ≡ J × B = µ1 u0 ⊥ + µ2 (b × u0 ),
(6)
where
µ1 = B 2 (σP + κi−1 σH ),
(7a)
µ2 = B 2 (σH − κi−1 σP ),
(7b)
and
u0 =(u0 , v 0 , w 0 )=(ui −un )ex +(vi −vn )ey +(wi −wn )ez
(8)
is the difference between the ion velocity and neutral wind.
Also, u0 ⊥ ≡u0 −(b·u0 )b is the perpendicular component of
the velocity difference projected onto a plane perpendicular
to the magnetic field: u0 ⊥ ·b=0. In Eq. (7), κi is the ratio of
the ion cyclotron frequency (ωci =eB/mi ) to the ion-neutral
collision frequency (νi ). In Eq. (8), ex , ey , and ez are the unit
vectors directed eastward (x), northward (y), and upward (z),
respectively. The subscripts i and n on the velocity components denote ion and neutral, respectively. Note that the second term in Eq. (6) is also perpendicular to b. Therefore, the
ion drag as parameterized by second expression in Eq. (6) is
perpendicular to b, which is consistent with its macroscopic
definition of J ×B. We also note that the coefficients µ1 and
µ2 in Eq. (6) are different from those shown in the literature
(e.g., Rees 1989, p. 206).
3316
X. Zhu et al.: A derivation of ion drag and Joule heating
Given E, B, un and other ionospheric parameters, one first
needs to solve Eq. (5) for ion velocity ui in order to derive
F I −D from Eq. (6) (see below and Appendix A for numerical
solutions for ui ). However, since F I −D is perpendicular to
b, it is possible to rewrite it in terms of the E-cross-B drift velocity uE ≡E×B/B 2 , which can be calculated directly from
E and B fields. In terms of uE Eq. (5) can be written as
uE = ui⊥ − κi−1 b × u0 ,
(9)
where ui⊥ denotes the corresponding perpendicular component of ui . Substituting Eq. (9) into Eq. (6) with some algebraic manipulations we arrive at a similar form for the ion
drag parameterization
F I −D = µ∗1 u00 ⊥ + µ∗2 (b × u00 ),
(10)
where u00 =uE −un is the velocity difference between the
E-cross-B drift velocity and neutral wind, u00 ⊥ is the corresponding perpendicular component: u00 ⊥ =u00 −(b·u00 )b.
Two coefficients in Eq. (10) are simply the first terms in
Eqs. (7a, b):
µ∗1
2
= σP B ,
(11a)
µ∗2 = σH B 2 .
(11b)
Equation (10) is the form often used for the atmospheric dynamics in the thermosphere (e.g., Dickinson et al., 1981).
It is worthwhile to compare the difference in their mathematical forms of ion drag between the microscopic form
δM s /δt in Eq. (1) and the parameterized form of Eqs. (6)
and (10). When there exists only one (dominant) kind of ion
that produces ion drag on the neutral wind, δM s /δt is proportional to u0 (=ui −un ) with a scalar coefficient (e.g., Schunk
and Nagy, 2000, p. 82). Therefore, the ion drag vector F I −D
in Eq. (1) is always parallel to the velocity difference vector
u0 . On the other hand, F I −D and u0 in the parameterized
forms of Eqs. (6) and (10) are no longer parallel unless u0 or
u00 is perpendicular to the magnetic field. The apparent paradox is caused by differing definitions of ion velocity. The ion
velocity in Eq. (1) represents the absolute movement of ions,
including the gyro-motion that defines the ion-cyclotron frequency and the ion-neutral collision frequency. On the other
hand, ui in Eq. (6) or uE in Eq. (10) is determined by a steady
state relationship such as Eq. (5) that represents the bulk ion
velocity, which has been averaged over a gyrocycle. It should
also be pointed out that the general macroscopic expression
for F I −D (=J ×B) on the right-hand side of Eq. (2) is universally correct by definition. Depending on whether J and
B are derived by time-dependent Maxwell equations or by a
steady state Eq. (5), J ×B can be reduced to either the microscopic form of Eq. (1) or the parameterized form of Eq. (6)
or Eq. (10).
It is also noted that below ∼200 km the second collision term on the right-hand side of Eq. (5) becomes nonnegligible (e.g., Kelley, 1989, Sect. 2.2). Therefore, it is
important to recognize the different definitions of u0 and u00
in Eqs. (6) and (10) because their coefficients are different
though the functional forms are very similar. In Fig. 1, we
show the Pedersen and Hall conductivities that are calculated from the NRLMSISE-00 model atmosphere (Picone et
al., 2002) and IRI ionospheric model (Bilitza et al., 1993) at
65◦ N magnetic latitude with the Ap index and the 10.7-cm
solar radio flux (in units of 10−22 W m−2 Hz−1 ) of 4.0 and
150.0, respectively. Results at midnight are provided in the
upper panel while those in the lower panel are at noon. Also
shown in the figure are the coefficients, µ1 and µ2 , that are
normalized by B 2 and used in Eq. (6) for the ion drag calculations. Figure 1 shows increasing differences with decreasing
altitude between µ1 and µ∗1 (=σP B 2 ) below ∼200 km due
to the contribution from the second term in Eq. (5). It also
shows significant differences between µ2 and µ∗2 (=σH B 2 )
because the second term in Eq. (7b) is greater than the first,
making µ2 negative. It can be shown that for κi−1 1 that
corresponds to a very low collision frequency, the two terms
in Eq. (7b) nearly cancel each other so that µ2 ≈0.
2.3 Parameterized ion drag in a dipole magnetic field
The Earth’s magnetic field in the thermosphere can be approximated by a magnetic dipole. To simplify the derivations
with clearer physics, we first assume coincident geographic
and geomagnetic poles. In this case, the unit vector of the
magnetic field, b, can be expressed as (e.g., Schunk and Nagy
2000, p. 314)
b = −(cos I )ey − (sin I )ez ,
(12)
where I is the dip angle between the magnetic field B and the
local horizontal direction. Substituting Eqs. (8) and (12) into
Eq. (6), we finally obtain the parameterized ion drag term
for the momentum equation of the neutral atmosphere in the
thermosphere
F I −D = ex , ey , ez

 
µ1
µ2 sin I −µ2 cos I
u0

2 −µ sin I cos I   v 0 
. (13)
 −µ2 sin I µ1 sin I 
1
0
2
w
µ2 cos I −µ1 sin I cos I
µ1 cos I
Note that the 3×3 matrix in Eq. (13) is neither symmetric
nor anti-symmetric. In most regions of the thermosphere, the
magnitudes of the horizontal velocities are far greater than
the magnitudes of the vertical ones. Therefore, the four terms
as sketched on the upper-left corner of Eq. (13) are often used
in the horizontal momentum equations for the atmospheric
dynamics in the thermosphere (e.g., Dickinson et al., 1981;
Rees 1989, p. 207). Because we have adopted the axialcentered dipole approximation (12) for the magnetic field,
the 2×2 sub-set of Eq. (13) for the horizontal ion drag happens to be anti-symmetric. Also note from Eq. (13) that the
major contribution to the ion drag in the vertical direction
will come from terms associated with the horizontal wind.
X. Zhu et al.: A derivation of ion drag and Joule heating
3317
65 MLAT, 00 MLT
400
350
z (km)
300
250
sigma_P
sigma_H
200
mu_1/B^2
- mu_2/B^2
(1E-4)*kappa_i
150
100
-9
10
10
-8
10
-7
-6
-5
10
10
Conductance (mho/m)
-4
10
-3
10
10
-3
10
-2
10
-2
65 MLAT, 12 MLT
400
350
z (km)
300
250
sigma_P
sigma_H
mu_1/B^2
- mu_2/B^2
(1E-4)*kappa_i
200
150
100
-8
10
10
-7
10
-6
10
-5
10
-4
10
-1
Conductance (mho/m)
Fig. 1. Pedersen and Hall conductivities (σP and σH , dashed lines) used in Eq. (10) as the coefficients for calculating ion drag and the
Figure 1
improved coefficients (µ1 /B 2 and −µ2 /B 2 , solid lines) in current parameterization of Eqs. (6) and (7) for ion drag. The thin dash-dotted
lines represent the ratio of the ion cyclotron frequency to the ion-neutral collision frequency scaled by 10−4 . The computations are done
based on the NRLMSISE-00 model atmosphere at 65◦ N magnetic latitude and the IRI90 ionospheric model. The Ap index and the 10.7-cm
solar radio flux (in units of 10−22 W m−2 Hz−1 ) used in computations are 4.0 and 150.0, respectively. The upper and lower panels are for
the midnight and noon magnetic local times, respectively.
1
For a displaced magnetic pole with declination angle δ, the
unit vector of the magnetic field is given by (e.g., Roble and
Dickinson, 1974)
µyy = µ1 (1 − cos2 δ cos2 I ),
(16b)
µzz = µ1 (1 − sin2 I ),
(16c)
b = ±(cos I sin δ)ex − (cos I cos δ)ey − (sin I )ez
µxy = ±µ1 sin δ cos δ cos2 I + µ2 sin I,
(16d)
µyx = ±µ1 sin δ cos δ cos2 I − µ2 sin I,
(16e)
µxz = ±µ1 sin δ sin I cos I − µ2 cos δ cos I,
(16f)
µzx = ±µ1 sin δ sin I cos I + µ2 cos δ cos I,
(16g)
µyz = −µ1 cos δ sin I cos I ∓ µ2 sin δ cos I,
(16h)
µzy = −µ1 cos δ sin I cos I ± µ2 sin δ cos I.
(16i)
,
(14)
where “+” and “–” are for the northern and southern geomagnetic hemispheres, respectively. The ion drag corresponding
to Eq. (14) is given by

 
µ
µ
µ
u0
xx
xy
xz

 0 
F I −D = ex , ey , ez  µyx µyy µyz  v
,
(15)
w0
µzx µzy µzz
where
µxx = µ1 (1 − sin2 δ cos2 I ),
(16a)
3318
Note that µyx 6=−µxy for a non-vanishing declination angle
δ. Therefore, in general, the ion drag tensor is not antisymmetric even for the simplified 2-D case for the horizontal
momentum equations.
3 Derivation of the parameterized Joule heating in the
energy equation
Under approach (i), the Joule heating is simply the second
moment of ion-neutral velocity difference in each species
equation (Schunk and Nagy 2000, p. 54). Similar to the
derivation of the parameterized ion drag in momentum equations we start from the energy equation of magnetohydrodynamics (approach (ii)) for the derivation of the parameterized
Joule heating (Schunk and Nagy, 2000, p. 196, p. 274; Rees,
1989, p. 124) for a bi-atomic gas (N2 and O2 )
D 5
5
p + p∇ ·u+τ : ∇u+∇ ·q = J ·(E +u×B), (17)
Dt 2
2
where p is the total atmospheric pressure, q is the total heat P
flow vector, τ is the total stress tensor, and
τ :∇u(= α,β ταβ [∂uα /∂xβ ]) denotes the double dot product of two tensors τ and ∇u. By use of an appropriate equation of state one can relate the pressure variation with the
temperature variation in the energy equation.
The terms on the left-hand side of Eq. (17) are again all
weighted by the density of both neutral species and charged
particles. On the other hand, the right-hand-side terms are
proportional to the density of charged particles only. Following the similar approximation for the momentum Eq. (3) we
arrive at the following energy equation for the neutral atmosphere
5
Dn 5
pn + pn ∇ · un + τ n : ∇un + ∇ · q n
Dt 2
2
= J ⊥ · (E ⊥ + un × B),
(18)
where the subscript n denotes the physical state for the neutral atmosphere. The right-hand side of Eq. (18) defines the
Joule heating for a neutral atmosphere because it is a thermal energy source that depends on electromagnetic fields
and electric currents. Note that we have also made two approximations on the right-hand side of Eq. (17) to arrive at
Eq. (18): (i) the total velocity on the second term has been
replaced by the neutral wind because charged particles are
considered minor tracers in the thermosphere; and (ii) the
first term J ·E has been approximated by J ⊥ ·E ⊥ because
the electric potential drop along the magnetic field is usually negligibly small in the thermosphere (e.g., Kelley, 1989,
p. 43). These two approximations have been made by the
basic rules of scale analysis described in the paragraph following Eq. (3). We will show below that making these two
approximations is also necessary in order to have a dynamic
system that is energetically consistent with the derivation of
ion drag above. In addition, these two approximations lead
to a non-negative Joule heating for the neutral atmosphere as
shown below in Eq. (19b). Therefore, the approach of scale
X. Zhu et al.: A derivation of ion drag and Joule heating
analysis used in simplifying equations is systematic and selfconsistent (e.g., Haltiner and Williams, 1980, Ch. 3).
Substituting Eqs. (4) and (5) into the right-hand side of
Eq. (18) we obtain the parameterized Joule heating in the
thermosphere
h
i
QJ ≡J ⊥ ·(E ⊥ +un ×B)=µ∗1 (1+κi−2 )|u0 |2 −(b·u0 )2 , (19a)
or in its explicitly non-negative form
h 2 i
2
QJ = µ∗1 u0 ⊥ + κi−2 u0 ,
(19b)
where µ∗1 is given by Eq. (11). Equation (19) indicates that
the Joule heating for the neutral atmosphere is proportional
to the Pederson conductivity, is independent of the Hall conductivity, and is non-negative.
By use of the relationship Eq. (9) between the bulk ion
velocity ui and the E-cross-B drift velocity uE we are also
able to derive QJ that contains the E-cross-B drift velocity
uE :
h
2 i
QJ = µ∗1 κi2 |uE − ui⊥ |2 + κi−2 u0 .
(20)
Unlike Eq. (10) for the ion drag parameterization, it is impossible to eliminate the explicit dependence of ui in QJ
after introducing uE . It is worth noting that both F I −D and
QJ vanish when the ion velocity coincides with the neutral
wind (ui =un ). Physically, this means that no momentum
and energy exchanges occur between ions and neutral particles as they are moving in the same direction at the same
speed. Such a physical argument is also explicitly shown in
the parameterized forms of Eqs. (6) and (19) for F I −D and
QJ , respectively. We will have both F I −D =0 and QJ =0 if
u0 vanishes. However, the E-cross-B drift velocity uE is different from the ion velocity ui if the second term in Eq. (5) or
Eq. (9) is non-negligible. For example, by definition, uE is
always perpendicular to both E and B whereas the collisions
between ions and neutral particles will drive the direction of
ui vector away from that of uE . As a result, a vanishing u0
will not generally lead to a vanishing u00 if the collision term
in Eq. (5) is included. Therefore, in general, one would not
expect to be able to transform an expression of F I −D or QJ
from a purely u0 -dependent form into a u00 -dependent one
without additional ui -dependent terms. The transform realized between Eqs. (6) and (10) happens to be a special case
since the parameterized F I −D is perpendicular to B and so
is the E-cross-B drift velocity uE .
From Eq. (20) we can find two limiting cases that the Joule
heating can be formally expressed in forms without ui terms:
(i) when the plasma becomes collisionless so that the second
term in Eq. (9) is negligible and the ion velocity coincides
with the E-cross-B drift velocity (uE =ui⊥ =ui ); (ii) when
the plasma becomes collision-dominant so that the ion velocity coincides with the neutral wind (ui =un ). Under these
assumptions, the Joule heating can be formally expressed as
2
QJ =µ∗1 κi−2 u0 = µ∗1 κi−2 |uE − un |2 as κi →∞ , (21)
QJ =µ∗1 κi2 |uE −ui⊥ |2 =µ∗1 κi2 |uE −un⊥ |2
as κi →0 . (22)
X. Zhu et al.: A derivation of ion drag and Joule heating
3319
From the vertical profile of κi (e.g., Kelley, 1989, p. 39)
one may expect Eqs. (21) and (22) to be good approximations in regions of high altitude, say >200 km, and low altitude, say <100 km, respectively. However, we should note
that the two assumptions used that led to Eqs. (21) and (22)
will lead to vanishing QJ . Therefore, one needs to know the
actual values of bulk ion velocity in order to have an appropriate evaluation of Joule heating. For reference we show in
Appendix A the explicit equations for solving ui components
in a dynamical model.
Explicit expressions for QJ by the velocity components
can be derived by substituting Eqs. (8) and (12) into Eq. (19)
to yield
h
i
QJ =µ∗1 (1+κi−2 )u02 +κi−2 (v 02 +w02 )+(v 0 sin I −w0 cos I )2 . (23)
Electromagnetic Energy
J⊥ •E⊥
Mechanical Energy
Kinetic
Energy
Thermal
Energy
J⊥ • (un × B)
For a displaced magnetic pole it can be shown that
h
QJ =µ∗1 (1+κi−2 − sin2 δ cos2 I )u02 +(1 + κi−2 − cos2 δ cos2 I )v 02
Fig. 2. Relationships of energy conversion among three types of
energy in the thermosphere.
+(cos2 I + κi−2 )w 02 ± 2 sin δ cos δ cos2 I u0 v 0
±2 sin δ sin I cos I u0 w 0 −2 cos δ sin I cos I v 0 w 0 .(24)
Eq. (26) represents the conversion of electromagnetic energy
into mechanical energy, which consists of both thermal and
kinetic energies for the atmosphere. This can be seen from
the sum of the right-hand sides of Eqs. (18) and (25) which
equals the negative of the right-hand side of Eq. (26).
By assuming the neutral atmosphere to be a major species
in the thermosphere we derived the parameterized ion drag
(Eq. 13) (or Eq. 15) and Joule heating (Eq. 23) (or Eq. 24),
which are functions of the velocity difference between the
bulk ion velocity and neutral wind. It can be explicitly shown
that the energetics of the parameterization remains consistent. In other words, the sum of the kinetic energy due to
ion drag and Joule heating for the neutral atmosphere as parameterized by the velocities and conductivities is equal to
the production (−J ⊥ ·E ⊥ >0) or the loss (−J ⊥ ·E ⊥ <0) of
electromagnetic energy when the same form of approximations Eqs. (4) and (5) for J ⊥ and E ⊥ are used. According
to Eq. (19b), the Joule heating under the current parameterization is positive unless the ion velocity coincides with
the neutral wind. Therefore, there will be an accompanying
conversion of kinetic energy into Joule heating when electromagnetic energy is generated by the atmospheric neutral
wind dynamo. Likewise, the electromagnetic energy cannot be fully converted into kinetic energy without producing
Joule heating in the thermosphere.
Joule heating as parameterized in Eq. (18) consists of two
terms
To conclude our derivations of F I −D and QJ parameterizations we emphasize that both the ion drag and Joule heating
are functions of the difference between the bulk ion velocity
and neutral wind. The dynamic effects of the charged particles on the neutral wind vanish if the ion velocity coincides
with the neutral wind (ui =un ).
4 Atmospheric energetics of the parameterized ion drag
and Joule heating
The total energy of the whole atmosphere under adiabatic
motion is conserved. From an energetics point of view, various terms in the momentum and thermal energy equations
represent conversions of energy from one form to another
(e.g., Dutton, 1986, Ch. 11; Holton, 1992, Sect. 10.4). Often, a simplification of the equations of motion for a specific problem requires that the energy conversion relations
still hold. To demonstrate the energy consistency of the system that includes charged particles we first note from Eq. (3)
that the induced change in kinetic energy density caused by
ion drag is
Dn 1 2
ρn
u
= un ·(J ×B) = −J ⊥ ·(un ×B).(25)
Dt 2 n I −D
For a system that includes currents and electromagnetic
fields, energy conservation is described by Poynting’s theorem (e.g., Jackson, 1975, p. 237; Thayer and Vickrey, 1992),
∂W
+ ∇ · S = −J · E ≈ −J ⊥ · E ⊥ ,
∂t
(26)
where W is the electromagnetic energy density, and
S=(E×B)/µ0 is the Poynting vector that represents the
electromagnetic energy flux density with µ0 being the permeability of free space. The term on the right-hand side of
J ⊥ ·E ⊥ =
σP |E ⊥ |2 −σP un ·(E ⊥ ×B)+σH (un ×b)·(E ⊥ ×B),
(27a)
J ⊥ · (un × B) =
σP |un ×B|2 −σP un ·(E ⊥ ×B)−σH (un ×b)·(E ⊥ ×B). (27b)
The first term (J ⊥ ·E ⊥ ), which also appears in Eq. (26) and
mathematically represents the conversion between electromagnetic energy and thermal energy, can be considered the
3320
X. Zhu et al.: A derivation of ion drag and Joule heating
conversion between electromagnetic energy and mechanical energy (Fig. 2). The second term (J ⊥ ·(un ×B)), which
also appears in Eq. (25) and mathematically represents the
conversion between thermal energy and kinetic energy, determines the partition between thermal energy and kinetic
energy within the mechanical energy. These two energetic
relationships as represented by two terms in Eq. (27) are
shown schematically in Fig. 2. When Eqs. (25), (18), and
(26) associated with all three types of energy (thermal, kinetic, and electromagnetic) are listed in parallel, the relationships among them become clear and self-consistent through
the conversion terms. These clear relationships suggest that
the second formulation of the ion drag and Joule heating in
terms of macroscopic Lorentz force and Ohmic dissipation,
Eqs. (3) and (18), is the best for understanding atmospheric
energetics. In addition, the two approximations used to derive the right-hand side of Eq. (18) from Eq. (17) become reasonable under the requirement of the energetics consistency
as described in Fig. 2. It provides a rationale for including
neutral wind un (rather than the average wind u in the primitive energy equation) in modifying the traditional definition
of Joule heating that consists only of σP |E ⊥ |2 contained in
J ⊥ ·E ⊥ .
Note that the last terms in Eqs. (27a) and (27b) cancel each
other. Therefore, Hall currents that are perpendicular to the
electric field do not contribute to Joule heating, which is also
explicitly shown in its more compact expression (Eq. 20).
However, on the basis of the above analysis, Hall currents
contribute to the conversions among three types of energy.
Adding Eqs. (27a) and (27b) and rearranging the terms, we
can rewrite Joule heating as (Lu et al., 1995)
QJ =QJ 1 +QJ 2 ≡
n
o
σP |E ⊥ |2 + σP |un ×B|2 −2σP un ·(E ⊥ ×B) ,
(28)
where the first term (QJ 1 ) is independent of the neutral wind
and is called “convection heating” (Lu et al., 1995) or “local
Joule heating” (Thayer, 1998). The terms in braces (QJ 2 ) are
called “wind heating” (Lu et al., 1995). Unlike expressions
(27a) and (27b), which emphasize the conversion among the
different types of energy, Eq. (28) emphasizes the effect of
neutral wind on Joule heating. Comparing Eq. (28) with
Eq. (19b) we can also express the convective heating in terms
of the ion velocity ui
h
i
σP |E ⊥ |2 = µ∗1 |ui⊥ |2 + κi−2 |ui |2 .
(29)
Because of the difficulty in directly measuring neutral wind
in the thermosphere (e.g., Kelley, 1989, p. 66), the contributions of the neutral wind to Joule heating were often neglected in earlier studies. As a result, neutral wind effects are
often neglected in textbook definitions of Joule heating. In
some textbooks, authors defined either σP |E ⊥ |2 (e.g., Kelley 1989, p. 270; Baumjohann and Treumann 1996, p. 88)
or J ·E (e.g., Kato 1980, p. 186) as Joule heating whereas in
others (e.g., Rees 1989, p. 127; Schunk and Nagy 2000, p.
402) authors only provided a descriptive role of Joule heating
in the thermal energy balance but gave no explicit expressions for its definition. Recent calculations have shown that
the magnitude of wind heating, QJ 2 , could be comparable to
that of convection heating, QJ 1 (e.g., Lu et al., 1995; Thayer
1998). Therefore, it is necessary to have a more precise and
self-consistent definition for Joule heating that includes the
effect of neutral wind.
In this paper, both ion drag and Joule heating have been
consistently derived from the general magnetohydrodynamic
Eqs. (2) and (18), respectively. An alternative approach of
deriving Joule heating including the neutral wind effect is
to simply replace E ⊥ in the traditional formula σP |E ⊥ |2
by E 0 ⊥ =E ⊥ +un ×B for the simple reasons (e.g., Jackson,
1975, p. 212): the ions are moving through the neutral gas
and it is the electric field that is measured in the reference
frame following the neutral wind that counts (e.g., Kivelson
and Russell, 1995, p. 494). However, under this argument,
the rest equations, such as the momentum equation or the
equation for the kinetic energy, also need to be revised into
the same moving frame while discussing the system energetics, which would make the study of the thermospheric dynamics more complicated.
The parameterized expressions of Eqs. (13) and (23) or
Eqs. (15) and (24) can be directly applied to thermospheric
general circulation models. The main advantages of adopting current set of parameterizations are: (i) both ion drag
and Joule heating are self-consistently dependent on the same
quantity of the difference between the bulk ion velocity and
neutral wind; (ii) the parameterizations are energetically conserved so changes in one type of energy in a dynamical model
will be exactly compensated by the changes in the rest two
types; (iii) the bulk ion velocity can be explicitly derived
from the modeled neutral wind (Appendix A) and can be
compared with the radar measurements in the thermosphere.
5
Concluding remarks
In this paper, we have derived ion drag term in the momentum equation and the Joule heating term in the thermal
energy equation for thermosphere atmospheric dynamics in
a consistent fashion, both being parameterized as functions
of the difference between the bulk ion velocity and neutral
wind. The parameterized expressions also depend on the
magnetic field, the Pedersen and Hall conductivities, and the
ratio of the ion cyclotron frequency to the ion-neutral collision frequency. Our derivation explicitly shows two different types of approximations in parallel: approach (ii) a
macroscopic formulation in terms of electrodynamic force
and Ohmic dissipation by Eqs. (3) and (18), respectively, and
approach (iii) a parameterized ion drag and Joule heating in
terms of bulk velocity difference by Eqs. (6) and (19), respectively. The atmospheric energetics was examined for both
sets of expressions. It is shown explicitly that Joule heating linearly depends on Pedersen conductivity and is always
positive unless the neutral wind coincides with the bulk ion
velocity and as a result of which both ion drag and Joule
X. Zhu et al.: A derivation of ion drag and Joule heating
heating vanish. The Hall currents contribute to the energetics conversion among thermal, kinetic, and electromagnetic
energies but have no effect on Joule heating.
We have also shown that there is an accompanying conversion of kinetic energy into Joule heating when electromagnetic energy is generated through the dynamo mechanism of
the atmospheric neutral wind. Likewise, electromagnetic energy cannot be fully converted into kinetic energy without
producing Joule heating in the thermosphere. The partition
between kinetic energy and Joule heating is analyzed by the
conversion terms. A similar analysis is also presented for the
effect of the neutral wind on Joule heating. When the selfconsistent and energetically conserved parameterizations are
applied to thermospheric general circulation models the bulk
ion velocity derived from the model neutral wind as shown
below in Appendix A can be compared with the radar measurements in the thermosphere.
Appendix A
Numerical solutions of the bulk ion velocity
Equation (9) that relates the E-cross-B drift velocity uE to
the bulk ion velocity ui and neutral wind un can be rewritten
as
κi [ui − (b · ui ) b] − b × ui = κi uE − b × un .
(A1)
Equation (A1) is a set of linear equations with respect
to ui components with the right-hand side being the force
terms. Assuming the general case of a displaced magnetic
pole Eq. (14) we can derive the matrix form of Eq. (A1) for
the bulk ion velocity components ui =[ui , vi , wi ]T
   

axx axy axz
ui
fx


(A2)
 ayx ayy ayz   vi  =  fy  ,
wi
fz
azx azy azz
where the coefficient matrix is similar to ion-drag matrix
shown in Eqs. (15) and (16) with µ1 and µ2 replaced by κi
and –1, respectively:
axx = κi (1 − sin2 δ cos2 I ),
(A3a)
ayy = κi (1 − cos2 δ cos2 I ),
(A3b)
azz = κi (1 − sin2 I ),
(A3c)
axy = ±κi sin δ cos δ cos2 I − sin I,
(A3d)
ayx = ±κi sin δ cos δ cos2 I + sin I,
(A3e)
axz = ±κi sin δ sin I cos I + cos δ cos I,
(A3f)
azx = ±κi sin δ sin I cos I − cos δ cos I,
(A3g)
ayz = −κi cos δ sin I cos I ± sin δ cos I,
(A3h)
azy = −κi cos δ sin I cos I ∓ sin δ cos I.
(A3i)
3321
The components of the right-hand-side force term are
given by
fx = κi uE − (sin I )vn + (cos δ cos I )wn ,
(A4a)
fy = κi vE ± (sin δ cos I )wn + (sin I )un ,
(A4b)
fz = κi wE − (cos δ cos I )un ∓ (sin δ cos I )vn .
(A4c)
To derive the horizontal components of bulk ion velocity
ui and vi , only the four terms as sketched on the upper-left
corner of Eq. (A2) are needed. A FORTRAN subroutine that
calculates the bulk ion velocity, ion drag and Joule heating is
available from X. Zhu ([email protected]) upon request.
Acknowledgements. X. Zhu thanks K. Liou, J. S. Saur, and
Y. Zhang for fruitful discussions and also thanks G. Lu, A. Richmond, and L. Paxton for insightful comments on the original
manuscript. The authors also thank two reviewers for making many
constructive comments, which has led to a significant improvement of the manuscript. This research was supported by the NASA
TIMED project under contract NAS5-97179 and in part by NASA
Grant NNG05GG57G and NSF Grant ATM-0091514 to the Johns
Hopkins University Applied Physics Laboratory.
Topical Editor U.-P. Hoppe thanks A. Aksnes and another
referee for their help in evaluating this paper.
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